2.1.

Overview

Our design starts with a dispersive x-ray spectrometer using gratings. Figure 1 shows a single-channel schematic design, with focusing optical assembly (“mirror”), gratings placed at some location along the optical axis, a multilayer (ML)-coated flat at the focus of the spectrometer thus created, and a detector for the reflected spectra. Figure 2 shows a rendered three-dimensional (3-D) view of all optical elements, with three independent channels and many grating facets for each channel. To have only one mirror but be able to measure at least three independent position angles and, consequently, three Stokes parameters, we subdivide the entrance aperture into six sectors at 60 deg apart. Using blazed gratings, where most power goes into $+1$ order, we align opposing sectors so that these focus to the same ML flat. There are then three spectroscopy channels, with an ML flat and a detector each. Finally, the period of the ML coating is laterally graded to match the Bragg peak of the ML reflectivity with the dispersion of the gratings.

Fig. 1

A schematic layout of a soft x-ray polarimeter. (Note, some dimensions and orientations are exaggerated for illustration.) (a) Overview of the optical elements with a focusing mirror at top, a transmission grating in the converging beam, an LGML coated flat at the spectroscopic focus, and an imaging detector at the prime focus (where zeroth order is observed). The gratings act like a prism, dispersing x-rays with short wavelengths closest to zeroth order and incident on the LGML at the small layer spacing end, whereas long wavelengths disperse farther from zeroth order, where the LGML period is largest. The optical axis is along the $z$-coordinate. Only one channel is shown here, which disperses along the $x$-axis. (b) Gratings on either side of the optical axis disperse the spectrum to a line in the focal plane (defining $z=0$) along the $x$-axis. Rays from gratings placed at different distances along $z$ have different incidence angles at the LGML and reflect to different $z$ values at the detector. (c) Approximate view from the point of view of the focusing optics. The LGML is shaded to indicate a changing period along its surface. Different sides of the telescope aperture disperse to different locations on the LGML, coinciding at the focal surface ($z=0$), along the center line of the LGML, and then reflecting to different locations on the detector (tilted slightly in this view).

Fig. 2

Raytrace-based schematic of the soft x-ray polarimeter design, as in the REDSoX polarimeter. In this sketch, the total length of the instrument is shortened relative to the mirror diameter to improve clarity. Photons enter through the annular aperture and are focused by the optics assembly made from concentric shells (gray cylinder). Gratings are arranged in three channels (shown in green, red, and yellow) comprising opposing 60-deg azimuthal sectors within the annular aperture. Monochromatic rays are colored according to the channel. The zeroth-order photons pass through the gratings to the central detector (blue). Others are diffracted and land on one of the three polarizing LGMLs (magenta), which reflect them onto that channel’s detector (blue). Viewers can interact with this figure in the Supplemental Content using Adobe Acrobat. (3D Multimedia 1, PDF, 5.66 MB) [URL: http://dx.doi.org/10.1117/1.JATIS.4.1.011005.1].

For an x-ray spectrometer designed to optimize spectral resolution along a single dispersion line in the focal plane, transmission gratings are usually arranged in a faceted Rowland configuration (A good example of an x-ray spectrometer in the Rowland configuration¹¹ is the High-Energy Transmission Grating Spectrometer¹² on the “Chandra” x-ray observatory.), with gratings tangent to a torus whose major and minor diameters are half of the distance from the grating assembly to the telescope’s focal surface. The basic design of our polarimeter involves matching the near-linear dispersion of the gratings to the lateral grading of a ML-coated flat set at about 45 deg to the incoming light. In the x-ray band, this orientation nulls the reflectivity of the polarization in the plane of the reflection (the $s$ polarization); at the Brewster angle (very close to 45 deg for x-rays) the reflected light is 100% polarized with the electric vector perpendicular to the reflection plane (the $p$ polarization). CCD detectors view each ML to determine the intensity reflected from that ML.

By orienting three ML-coated flats at position angles 120 deg apart, we can measure the three Stokes parameters $I$, $Q$, and $U$ at any time without instrument rotation. Each channel measures a specific combination of $I$, $Q$, and $U$; arranging three flats to be oriented at 120 deg from each other provides optimal independence among the measurements so that the three Stokes parameters can be determined. The fraction of linearly polarized light is given by $\mathrm{\Pi }=\sqrt{Q^2+U^2}/I$. The sensitivity of an x-ray polarimeter is often characterized by the minimum detectable polarization (MDP) that can be achieved for a given source, as discussed in Sec. 3.6. The zeroth-order CCD detector can also be used to determine $I$, providing some redundancy. See Sec. 3.6 and particularly Eq. (18) for details about how to compute the Stokes parameters from the observations.

To get an approximate, intuitive feel for a workable design, we begin with a simplistic model, where gratings are placed in the converging beam of a focusing mirror with focal length $F$. We constrain the design for simplicity of fabrication to require using gratings with all the same period, $P$. Furthermore, the system must focus x-rays of a given wavelength, $\lambda$, to a relatively small region on the ML, where the Bragg condition is satisfied, resulting in placing the ML-coated flat (not the detector) at the spectroscopic focus of the system. We define a coordinate system where $z$ is oriented along the optical axis, with $z=0$ at the on-axis focus and the $+z$-direction is oriented toward the mirror. A transmission grating is placed at location $(0,y,z)$, with its normal along the direction to the telescope focus at (0,0,0) but dispersing x-rays to positions $(x,\mathrm{0,0})$. The dispersion of the grating is given by the grating equation $m\lambda =P\sin \alpha$, where $m$ is the grating order of interest (which we take to be $+1$) and $\alpha$ is the dispersion angle. For small angles, $\alpha \approx x/z$. Our goal is to match the dispersed wavelength to the peak reflectivity of a laterally graded multilayer (LGML), for which the Bragg condition is given by $\lambda =2D\cos \theta$, where $D$ is the ML period and $\theta$ is the angle of incidence relative to the LGML normal; $\theta \approx y/z+\pi /4$, for a ML-coated flat that is placed at 45 deg to the optical axis. For a linear period gradation, $D=Gx$, where $x$ is the distance along the LGML. Again, the simplest design is when $G$ is the same for all LGMLs.

We match the LGML’s Bragg peak to the grating dispersion by setting $P\sin \alpha \approx Px/z=2D\cos \theta =2Gx\cos \theta$, giving $z=P/(2G\cos \theta )$, so $z=P/(2G\cos [y/z+\pi /4])$. Thus, gratings are shifted along the $z$-axis as the grating is displaced away from the optical axis along the $y$-axis. Because $y/z$ is usually small compared with $\pi /4$, $z$ varies with $y$ nearly linearly. Next, we provide a rigorous derivation of how to place gratings in our design.

2.2.

Grating Positioning

Determining the positions of the gratings is the key to our design. Here, we provide a rigorous derivation of the requirements for a multigrating design. In what follows, we use two approximations: $P\gg \lambda$ and $\alpha \ll 1$. For the implementation discussed in Sec. 3, $P=2000Å$ while $\lambda <75Å$, so $\lambda /P<0.04$. The maximum value of $\alpha$ is dictated by realistic x-ray optics for broadband use with two reflections, so $\alpha =4{\alpha }_{\mathrm{g}}$, where ${\alpha }_{\mathrm{g}}$ is the graze angle for total internal reflection that yields reflectivities above 90%. For our proposed implementation, ${\alpha }_{\mathrm{g}}<1.3\mathrm{deg}$, so $\alpha <0.1\mathrm{rad}$. Both approximations are justified post facto as the design is verified by ray tracing, as shown in Sec. 4.

First order from the grating will fulfill the local Bragg condition depending on the angle between the ray and LGML normal $\overrightarrow{n_m}$. Photons pass through the mirror and are focused toward (0,0,0). We choose the $+x$-axis of the coordinate system as the dispersion direction. The $+x$-axis is also chosen to be in the plane of the LGML, whose unit normal pointing toward the reflective surface is

The ML coating is laterally graded, i.e., the period depends on the $x$-coordinate

2.2.2.

General grating positioning

We now introduce a spherical coordinate system as shown in Fig. 3. The polar angle (${\gamma }_g$) is measured with respect to the $x$-axis and the $yz$-plane is the equatorial plane. This frame is more convenient than the usual definition, where the polar angle is measured with respect to the $z$-axis because $x$ is the direction along which the period of the ML flat changes. Consider a grating located at

Eq. (8)

$$\overrightarrow{X_g}=r_g\left(\begin{array}{c}\cos {\gamma }_g\\ \sin {\gamma }_g\sin {\beta }_g\\ \sin {\gamma }_g\cos {\beta }_g\end{array}\right),$$

where $r_g$ is the distance between the grating and the focal point (located at the origin of the coordinate system), ${\gamma }_g$ is the polar angle, and ${\beta }_g$ is the azimuthal angle measured in the $yz$-plane from the $z$-axis. These angles are shown in Fig. 3. A photon incident upon this grating will have the direction vector $\overrightarrow{X}=-(\sin {\gamma }_g,\cos {\gamma }_g\sin {\beta }_g,\cos {\gamma }_g\cos {\beta }_g)$. We place the gratings so that they are essentially perpendicular to the incoming photons (except for an adjustment for the blaze angle described in Sec. 2.5) to reduce the shadows cast by the grating support structure (see Sec. 3). We also want to orient the gratings such that they disperse along the $x$-axis. We can then write the diffraction of photons as a rotation with angle $\alpha$ from Eq. (4) around the axis $\overrightarrow{A}$

Eq. (9)

$$\overrightarrow{A}=\overrightarrow{X_g}\times \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ \sin {\gamma }_g\cos {\beta }_g\\ -\sin {\gamma }_g\sin {\beta }_g\end{array}\right).$$

Fig. 3

Sketch of the geometry, showing three polarimetry channel detectors (blue), one direct imaging detector (blue rectangle at the coordinate system origin), three ML flats (magenta rectangles), and three gratings (red rectangles). The distance $r_g$ (orange vector) is from the origin to the center of the grating marked by the gray point. The angle ${\gamma }_g$, indicated by the blue wedge, is the angle to the grating from the $x$-axis. The angle ${\beta }_g$, indicated by the dark gray wedge, is the angle between the $z$-axis and the projection of the line to the grating center onto the $yz$-plane.

We can write the rotation matrix around this axis as

Eq. (10)

$$R=\cos \alpha \mathsf{E}+\sin \alpha \left(\begin{array}{ccc}0& \sin {\gamma }_g\sin {\beta }_g& \sin {\gamma }_g\cos {\beta }_g\\ -\sin {\gamma }_g\sin {\beta }_g& 0& 0\\ -\sin {\gamma }_g\cos {\beta }_g& 0& 0\end{array}\right)+(1-\cos \alpha )\left(\begin{array}{ccc}0& 0& 0\\ 0& \dots & \dots \\ 0& \dots & \dots \end{array}\right),$$

where $\mathsf{E}$ is the identity matrix and we use the approximation $\cos ({\alpha }_g)\approx 1$. The direction vector $\overrightarrow{X_1}$ of a dispersed first-order photon is

Eq. (11)

$$\overrightarrow{X_1}=R\overrightarrow{X}=-\left(\begin{array}{c}\sin {\gamma }_g\\ \cos {\gamma }_g\sin {\beta }_g\\ \cos {\gamma }_g\cos {\beta }_g\end{array}\right)-\frac{\lambda }{P}\left(\begin{array}{c}{\cos }^2{\gamma }_g\\ -\sin {\gamma }_g\cos {\gamma }_g\sin {\beta }_g\\ -\sin {\gamma }_g\cos {\gamma }_g\cos {\beta }_g\end{array}\right),$$

where we have used Eq. (4).

We can now plug this photon direction vector into the Bragg condition, Eq. (3), use Eq. (2), and set $G_0=0$ to obtain

Eq. (12)

$$\lambda =\sqrt{2}Gx\cos {\gamma }_g(\sin {\beta }_g+\cos {\beta }_g)(1-\frac{\lambda }{P}\sin {\gamma }_g),$$

where the last term can be dropped because the wavelength $\lambda$ is much smaller than the grating constant $P$ in our implementation.

We need to express the position $x$ in terms of the grating coordinates and the photon wavelength $\lambda$. The equation for a diffracted ray is

Eq. (13)

$${\overrightarrow{X}}^{\prime }=\overrightarrow{X_g}+c\overrightarrow{X_1},$$

for some value of $c$. It is sufficient to just write out the $z$-component of this equation to see where the ray intersects the LGML, which will happen when the ray passes the plane $z=0$. Solving for $c$ gives

Eq. (14)

$$c=\frac{r_{g}P}{P-\lambda \sin {\gamma }_g}\approx r_g,$$

and substituting this result into the $x$-component of Eq. (13) yields

Eq. (15)

$$x=\frac{r_g}{P}\lambda {\sin }^2{\gamma }_g.$$

Last, we obtain $r_g$ for a grating for any given ${\gamma }_g$, ${\beta }_g$ by inserting Eq. (15) into Eq. (12)

Eq. (16)

$$r_g=\frac{P}{\sqrt{2}G{\sin }^3{\gamma }_g(\sin {\beta }_g+\cos {\beta }_g)}.$$

With Eq. (16), we then have the position of the grating using Eq. (8) given values of ${\gamma }_g$ and ${\beta }_g$.

2.3.

Filling the Space Available With Gratings

Equation 16 specifies where a grating must be positioned for a given $({\gamma }_g,{\beta }_g)$ to direct photons with $m=+1$ to the LGML so that the Bragg condition is satisfied. The transmission gratings planned for our implementation of the design have finite size and should be placed normal to the incoming (converging) photons. For these reasons, the gratings cannot follow the shape of the surface given by Eq. (16) exactly but can be approximated sufficiently. In our implementation, we place gratings in a rectangular grid to fill the annulus under the mirror shells that is traversed by the photons after focusing. We apply Eq. (16) to the center of each grating and use ray-tracing to calculate the nonideal effects arising from the finite size of flat gratings. The result for our sounding rocket implementation can be seen in Sec. 3.2.

2.4.

Multiple Channels

As described in Secs. 1 and 2.1, the REDSoX polarimeter consists of three LGMLs and corresponding detectors that measure polarization independently. To achieve this, the rectangular grid of gratings does not cover the full annulus but only two opposing sectors, each of which is 60-deg wide. An opposing pair of sector images onto one of the LGMLs; the combination of gratings, LGML, and detector comprises a “channel.” One of the two segments in each channel is “high” [larger $r_g$ because $\beta >0$, see Eq. (16)], and the other one is “low” (smaller $r_g$ because $\beta <0$).

There are three channels. For channels 2 and 3, all gratings and the corresponding LGML and CCD detector are rotated by 120 deg and $-120\mathrm{deg}$, respectively, around the optical axis with respect to channel 1. See Figs. 2, 8, and 13.

2.5.

Grating Blaze Angle

The grating placement as discussed above is for transmission gratings with similar efficiencies in positive and negative diffraction orders. To reduce mass and cost, we plan to use gratings in which the diffraction efficiency is maximized to one side by blazing (tilting) the grating surface. To take advantage of the blaze feature, every grating is rotated by ${\theta }_B$ around the grating bar axis, which we will define as its pitch (Fig. 4). Generally, ${\theta }_B$ is small enough that there is no significant shadowing by the grating support structure.

Fig. 4

Sketch of a critical angle transmission (CAT) grating. The solid blue line is the path of the incoming photon. The red line is perpendicular to the incoming photon and would be the plane of a traditional transmission grating (gray grating bars). Instead, the CAT grating is blazed by the angle ${\mathrm{\Theta }}_B$, which makes photons “bounce off” the walls of the grating bars (thick black lines) and all the dispersed signal ends up toward the right of the zeroth order. Instead, the blaze angle was $-{\mathrm{\Theta }}_B$, photons would “bounce off” the other side of the grating bars and would be blazed toward a location to the left of the zeroth order. Note that for the blaze angle ${\mathrm{\Theta }}_B$ the angle between the incoming ray (solid blue line) and the diffracted ray (dashed blue line) is $2{\mathrm{\Theta }}_B$. Sometimes this $2{\mathrm{\Theta }}_B$ angle is called “blaze angle” in the literature, which can lead to confusion.

3.1.

Focusing Optics

The broadband focusing mirror should work well in the energy range appropriate to the available MLs that can be used in the 0.1- to 1.0-keV range. Type I Wolter designs are effective here, so we define the inner and outer radii as $r_I$ and $r_O$. The telescope’s $f$-ratio should not be very small; the sounding rocket constraints provide an $f$-ratio over 5, which suffices. The requirement on the mirror half-power diameter (HPD) will be determined by setting a limit on the spectrometer’s spectral resolution, which is derived from the gradient, $G$, of the laterally graded ML (see Sec. 3.3). We anticipate that HPDs of $<{30}^{″}$ will be sufficient and have been demonstrated by researchers at the Brera Observatory^14,15 and at the Marshall Space Flight Center.¹⁶ Five mandrels were made for the Micro-X project¹⁷ with a 2500-mm focal length, so we adopt this focal length. The focal plane scale is ${82}^{″}/\mathrm{mm}$.

For the REDSoX polarimeter, there would be nine shells, with inner radii ranging from 14.96 to 20.22 cm and outer radii ranging from 16.94 to 22.88 cm. The mirror geometric area is about $640{\mathrm{cm}}^2$, and we assume that the support structure blocks 12% of this area and then use reflectivity given by two reflections from Ni-coated surfaces using the known graze angles. The total mass of the mirror is expected to be about 78 kg.

3.2.

Gratings

For gratings, we baseline critical angle transmission (CAT) gratings¹⁸ developed in the Space Nanotechnology Lab (SNL) at MIT. These can now be reliably produced in a $10\times 30\mathrm{mm}$ (and a $30\times 30$) format with period $P=2000Å$ on 0.5-mm-thick Si wafer substrates.^19,20 We extrapolated measurements taken at the advanced light source (ALS) to 70 Å using an efficiency model and then accounted for structural obscuration to generate an efficiency model for our baseline effective area. We further note that the measured efficiencies in the 30- to 70-Å range are generally closer to the models than the measurements below 30 Å.²¹

A grating in a holder is shown schematically in Fig. 6 and several gratings are assembled stair-stepped into one “sector” as shown in Fig. 7. The sectors are mounted approximately azimuthally about the optical axis, as shown in Fig. 8, so that the dispersion direction is along the long dimension. The gratings are mounted in a blazed configuration, rotated so that x-rays reflect off of the polished grating sidewalls, increasing the efficiency in (what we define as) positive orders. The blaze condition is described in Sec. 2.5 and shown in Fig. 4. The system MDP (from Sec. 3.6) varies inversely with the square root of the grating efficiency, so we computed the MDP as a function of blaze angle, finding a minimum at about 0.8 deg (see Fig. 9). To increase the efficiency for dispersed x-rays, the throughput in the zeroth order is decreased, as shown in Fig. 9.

Fig. 6

Engineering drawing of a CAT grating in a holder. The L2 support structure is visible as the hexagonal structure that is 0.5-mm deep; the grating is a $4\text{-}\mu \mathrm{m}$ layer on top, with the dispersion along the long axis.

Fig. 7

Engineering design of an assembly of gratings for one (high) sector of the REDSoX polarimeter. The gratings are aligned to disperse either to the left or to the right, depending on whether the assembly is high or low along the optical axis. The gratings are stair-stepped so that the Bragg condition is met at the ML. Ribs have pins to securely position gratings below the Invar mounts and flexible arms above to maintain location.

Fig. 8

Grating assemblies as installed on the optical bench of the REDSoX polarimeter.

Fig. 9

For the example of Mk 421, this shows the calculation of (a) the MDP and (b) zeroth-order count rate as a function of the CAT grating blaze angle in the REDSoX polarimeter implementation. At 0.8 deg, the MDP is optimized at the expense of the count rate in the zeroth order. The MDP is insensitive to the blaze angle within 0.15 deg of the optimum, setting the mounting angle tolerance.

3.3.

Laterally Graded Multilayer Coatings

Under NASA funding, there is demonstrated performance of LGMLs coated flats. An image of such an LGML is shown in Fig. 10. LGMLs have been made by reflective x-ray optics (RXO) and the center for x-ray optics (CXRO) that are suitable for our design. The coating substrates are portions of highly polished Si wafers, about 0.5-mm thick. They have linear period spacing gradients of $0.88Å/\mathrm{mm}$ to reflect and polarize x-rays from 17 to 75 Å (165 to 730 eV). The coating material pairs were $\mathrm{W}/{\mathrm{B}}_4\mathrm{C}$, C/CrCo, $\mathrm{Cr}/{\mathrm{B}}_4\mathrm{C}/\mathrm{Sc}$, and $\mathrm{La}/{\mathrm{B}}_4\mathrm{C}$. Reflection efficiencies at 45 deg were measured at the ALS in Berkeley, California, at 1- or 2-mm spacing for each LGML.^9,10 Results of these tests are shown in Fig. 11 for the wavelength range of interest. Across each LGML, the Bragg peak deviates from linearity by $<1\%$, $\times 2$ smaller than the FWHM of the peak (Fig. 12).

Fig. 10

A LGML coated flat from RXO in a lab holder. It is about 0.5-mm thick and 47-mm long. The ML period, $d$, increases from left to right.

Fig. 11

Reflectivity measurements of LGMLs^9,10 using a partially polarized beam at the ALS in Berkeley. The reflectivities were sampled at 1-mm spacing, starting below 30 Å (about 0.4 keV) for the Cr/Sc sample (green) and at 2-mm spacing for the C/CoCr (red) and $\mathrm{La}/{\mathrm{B}}_4\mathrm{C}$ (violet) samples.

Fig. 12

Bragg peak wavelength as a function of position along a LGML (A13164), made from alternating layers of C and a CoCr alloy. The centroid of the Bragg peak is linear with position along the LGML to better than 1%, sufficient for an instrument, such as the REDSoX polarimeter.

The reflectivities of the LGMLs are critical to the performance of this design. The baseline implementation selects the 35 to 75 Å wavelength range due to the extents of the CCD detectors and the spectrometer dispersion relation. Referring to Fig. 11, the Cr/Sc LGML has the best reflectivity in the 35 to 50 Å range, C/CrCo is best in the 40 to 67 Å range, and $\mathrm{La}/{\mathrm{B}}_4\mathrm{C}$ is best longward of 67 Å. The flight LGML will be a mosaic of the three types, cut precisely at positions corresponding to Bragg peaks at 50 and/or 67 Å depending on the LGML and mounted as an assembly for flight. The LGML assemblies will be the same for each channel. When corrected for the ALS beam polarization, the average reflectivity to unpolarized light is about 10%.

3.4.

Detectors

The focal plane layout is shown in Fig. 13. For the CCD detectors, we use commercially available CCD cameras from XCAM. The CCDs are semicustom devices manufactured by e2v technologies. These CCDs have typical quantum efficiencies of 80% to 90% in the energy range of 150 to 500 eV. Each CCD has a $26.11\text{-}\times 25.73\text{-}\mathrm{mm}$ image area with 1632 horizontal and 1608 vertical active pixels and full frame readout. The small pixel size of $16\mu \mathrm{m}\times 16\mu \mathrm{m}$ allows high-resolution imaging, but we can sum rows for fast readouts. Used with the spectrometer dispersion along the diagonal, the useful dispersion range is about 35 mm, giving a 35 to 75 Å (165 to 350 eV) bandpass.

Fig. 13

Interior of the detector vacuum chamber showing almost all focal plane components: LGMLs, CCD faces, and flex prints (gold). A black anodized CCD baffle with a horizontal rectangular aperture restricts the view of the CCDs to its LGML. The cylinder at the base of the focal plane will be filled with liquid ${\mathrm{N}}_2$ prior to launch in the sounding rocket implementation.

3.5.

Background

To estimate the particle background, we use results from the Suzaku satellite in low Earth orbit. For the REDSoX polarimeter, launching either from White Sands, Minnesota, or from Australia, the maximum altitude will be somewhat lower than Suzaku, so the Suzaku background will be taken as an upper limit. The particle background in the Suzaku backside-illuminated CCD was $5\times {10}^{-8}\mathrm{cnt}/\mathrm{s}/\mathrm{keV}/\text{pixel}$ or $1.5\times {10}^{-5}\mathrm{cnt}/\mathrm{s}/{\mathrm{mm}}^2$ in a 0.2-keV band at 0.3 keV.²² In the cross-dispersion direction, the dispersed spectral extraction region will be about twice the size of the optics HPD, about 60″ or 0.7-mm wide. In the dispersion direction, the spectra are 35-mm long. Using three detectors and exposing for 300 s yields 0.3 counts expected from particles. The x-ray background in the REDSoX polarimeter bandpass is dominated by galactic emission, about ${10}^{-3}\mathrm{cnt}/\mathrm{s}/{(\mathrm{arcmin})}^2$ in the ROSAT C band or $540\mathrm{ph}/{\mathrm{cm}}^2/\mathrm{s}/\mathrm{sr}/\mathrm{keV}$ for ROSAT’s area ($220{\mathrm{cm}}^2$) and bandwidth (0.1 keV).²³ Thus, we expect about 30 cnt in 300 s across the REDSoX polarimeter band in the solid angle of each detector. The LGMLs, however, have a bandpass of $\delta E=0.025E$, giving $<2\text{counts}$ expected across the three detectors. Compared with a source with 50 to 1000 counts, these backgrounds are negligible. To avoid solar scattered background, we require that the angle between the telescope’s line of sight and the Sun should be $>120\mathrm{deg}$, as demonstrated by ROSAT.²³

3.6.

Baseline Performance

For a polarimeter, a figure of merit is the MDP at 99% confidence, given as a fraction $\mathrm{MDP}=4.29\sqrt{(R+B)/T}/(\mu R)$, where $R$ is the source count rate, $B$ is the background count rate in a region that is the size of a typical image, $T$ is the observation time, and $\mu$ is the modulation factor of the signal relative to the average signal for a source that is 100% polarized.^4,24 For most polarimeters operating in the 2- to 8-keV band, the modulations factors peak at 0.5 to 0.7 [e.g., 3], whereas for the REDSoX polarimeter and other polarimeters based on Bragg reflection, $\mu >0.90$. In Sec. 4, we verify that $\mu$ is this high for the REDSoX polarimeter.

The effective area of the system was also validated using ray tracing developed for the REDSoX polarimeter project (see Sec. 4). The integrated area of the system, defined as $\mathcal{A}=\int A(\lambda )\mathrm{d}\lambda$ is $185{\mathrm{cm}}^2Å$. For a source with a flat spectrum, $R=I\mathcal{A}$, where $I$, one of the Stokes parameters, is the source flux in ph ${\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}{Å}^{-1}$. The zeroth-order detector also provides a measurement of $I$, with ${\mathcal{A}}_0=488{\mathrm{cm}}^2⏧$ but with a less definite energy band due to the CCD spectral resolution.

Briefly, we describe how to measure the Stokes $Q$ and $U$ parameters using just the polarimetry channel data, for the simple case of a flat input spectrum. In flight, the system would measure count rates in each of three channels, denoted by $R_i$, for $i=1\dots 3$. For given values of $I$, $Q$, and $U$